7.03 Loans
Banks and other financial institutions allow us to borrow money in the form of a loan, but in return they charge interest on the balance of the loan. While being charged compound interest on the balance of the loan, you also make regular repayments to reduce this balance and eventually repay the loan entirely. So we call these reducing balance loans, and as the amount owing on the loan gets smaller after repayments are made, so does the amount of interest that is charged each period.
Deposit: an initial, partial payment for the cost of the loan.
Balance: the amount owing after interest has been incurred or repayments have been made.
Repayment: a fixed amount paid at regular time periods to repay the balance and any interest incurred.
Interest: an amount charged for loaning you the amount of the balance. Usually calculated as a percentage of the balancing owning using the compound interest formula.
Interest rate: the amount of interest charged expressed as a percentage.
Analysing a reducing balance loan with a table
A table is a useful tool to analyse reducing balance loans. Loan tables can display the progression of the of the loan balance and the interest to be paid in each period. We can also use these figures to calculate the total cost of a loan and the total interest paid.
Exploration
Let's consider the example of using a loan of $\$1000$$1000 at an interest rate of $2.84%$2.84% p.a. compounded annually, that is, interest is charged on the balance of the loan once a year. The loan is for a term of $4$4 years with repayments of $\$268$$268 made at the end of each year.
The interest charged in the first year:
| $\text{Interest}$Interest | $=$= | $0.0284\times\$1000$0.0284×$1000 |
$2.84%$2.84% of the initial loan |
| $=$= | $\$28.40$$28.40 |
|
Balance after $1$1 year:
| $\text{Balance at end of year}$Balance at end of year | $=$= | $\text{Balance at start of year}+\text{Interest}-\text{Payment}$Balance at start of year+Interest−Payment |
| $=$= | $\$1000+\$28.40-\$268$$1000+$28.40−$268 | |
| $=$= | $\$760.40$$760.40 |
This becomes the balance for the start of the second year. And we can continue to find the interest paid for the year by calculating $2.84%$2.84% of the current balance and also the balance at the end of the year using:
$$
The reducing balance for the $4$4 year term of the loan is shown in the table below.
| Year | Balance start of year | Interest added this period | Repayment | Balance end of year |
|---|---|---|---|---|
| $1$1 | $\$1000.00$$1000.00 | $\$28.40$$28.40 | $\$268$$268 | $\$760.40$$760.40 |
| $2$2 | $\$760.40$$760.40 | $\$21.60$$21.60 | $\$268$$268 | $\$514.00$$514.00 |
| $3$3 | $\$514.00$$514.00 | $\$14.60$$14.60 | $\$268$$268 | $\$260.60$$260.60 |
| $4$4 | $\$260.60$$260.60 | $\$7.40$$7.40 | $\$268$$268 | $\$0$$0 |
From the table we can see both the interest to be paid and balance reduce over the term of the loan.
We can calculate the total cost of the loan(total loan amount) using the fact that there were $4$4 equal payments of $\$268$$268.
| $\text{Total loan amount}$Total loan amount | $=$= | $\text{Number of repayments}\times\text{Loan repayment}$Number of repayments×Loan repayment |
| $=$= | $4\times\$268$4×$268 | |
| $=$= | $\$1072$$1072 |
We can calculate the total interest paid by summing the amounts in the interest column of the table or by finding the difference between the total cost of the loan and the initial amount borrowed.
| $\text{Total interest paid}$Total interest paid | $=$= | $\text{Total loan amount}-\text{Initial amount borrowed}$Total loan amount−Initial amount borrowed |
| $=$= | $\$1072-\$1000$$1072−$1000 | |
| $=$= | $\$72$$72 |
Hence, this loan cost a total of $\$1072$$1072, of which $\$72$$72 was interest.
$\text{Total loan amount}=\text{Number of repayments}\times\text{Loan repayment}$Total loan amount=Number of repayments×Loan repayment
$\text{Total interest paid}=\text{Total loan amount}-\text{Initial amount borrowed}$Total interest paid=Total loan amount−Initial amount borrowed
Final payments
In the example above the payments coincided exactly with paying the loan off in $4$4 years. In practice we often round payments to a convenient figure such as rounding a home loan repayment to the nearest $\$10$$10. This leads to the final payment being different from the previous payments. The final adjusted payment is to pay off the remaining balance in the last period plus interest on this amount.
Worked example
Example 1
A small in store loan to is used to purchase a new pair of shoes for $\$300$$300. Interest of $10%$10% per week is charged on the balance of the loan, and we are able to pay back $\$100$$100 each week.
(a) Using the table below, find the final payment due on the loan.
| Period (n) | Value at beginning of period | Interest added this period | Repayment | Value at end of period |
|---|---|---|---|---|
| $1$1 | $\$300$$300 | $\$30$$30 | $\$100$$100 | $\$230$$230 |
| $2$2 | $\$230$$230 | $\$23$$23 | $\$100$$100 | $\$153$$153 |
| $3$3 | $\$153$$153 | $\$15.30$$15.30 | $\$100$$100 | $\$68.30$$68.30 |
| $4$4 | $\$68.30$$68.30 | $\editable{}$ | $\editable{}$ | $\$0$$0 |
Think: The final payment needs to pay off the remaining balance of $\$68.30$$68.30 plus interest on this amount.
Do:
| $\text{Interest}$Interest | $=$= | $\$68.30\times0.1$$68.30×0.1 |
| $=$= | $\$6.83$$6.83 |
| $\text{Final payment}$Final payment | $=$= | $\text{Remaining balance}+\text{Interest}$Remaining balance+Interest |
| $=$= | $\$68.30+\$6.83$$68.30+$6.83 | |
| $=$= | $\$75.13$$75.13 |
(b) Find the total cost of the loan.
Think: There were three full payments of $\$100$$100 plus a final adjusted payment of $\$75.13$$75.13.
Do:
| $\text{Total loan amount}$Total loan amount | $=$= | $\text{Number of full payments}\times\text{Repayment}+\text{Final adjusted payment}$Number of full payments×Repayment+Final adjusted payment |
| $=$= | $3\times\$100+\$75.13$3×$100+$75.13 | |
| $=$= | $\$375.13$$375.13 |
(c) How much interest was paid on the loan?
Think: The interest is the difference between the total cost of the loan and the amount borrowed.
Do:
| $\text{Total interest paid}$Total interest paid | $=$= | $\text{Total loan amount}-\text{Initial amount borrowed}$Total loan amount−Initial amount borrowed |
| $=$= | $\$375.13-\$300$$375.13−$300 | |
| $=$= | $\$75.13$$75.13 |
Practice questions
Question 1
Dylan takes out a loan to purchase a property. He makes equal monthly loan repayments of $\$4600$$4600 over $27$27 years to pay it off. The interest of $8%$8% is compounded annually.
What is the total loan amount?
Question 2
Ivan takes out a car loan for $\$24000$$24000. He is charged $8.1%$8.1% per annum interest, compounded monthly. Ivan makes repayments of $\$450$$450 at the end of each month.
Complete the values in the empty cells in the table below. Give your answers correct to the nearest cent.
Month Opening Balance Interest Repayment Closing Balance 1 $24000$24000 $162.00$162.00 $450$450 $23712.00$23712.00 2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Question 3
You take out a personal loan of $\$10000$$10000 at $11%$11% reducible p.a. The term of the loan is $3$3 years, and yearly repayments of $\$2600$$2600 are made. The balance owing is paid at the end of $3$3 years.
Complete the loan repayment table:
Year Balance of loan at beginning of period Interest charged during period Repayment Balance of loan at end of period $1$1 $10000$10000 $\editable{}$ $2600$2600 $\editable{}$ $2$2 $\editable{}$ $\editable{}$ $2600$2600 $\editable{}$ $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ What is the total amount of interest charged on the loan?
What is the total repayment over $3$3 years?